Question: Multiply. $0.48 \times 23.9=$
Answer: $48 \times 239$ and $0.48\times 23.9$ multiply the same digits in the same order. So, the product of both problems will also have the same digits in the same order. Let's multiply $48 \times 239$. Then we can estimate to place the decimal point in the product of $0.48 \times 23.9$. $\begin{aligned} 23{9}&\\ \underline{ \times 4{8}}&\\ {72}& {8} \times {9\text{ ones}}\\ \end{aligned}$ $\begin{aligned} 2{3}9&\\ \underline{ \times 4{8}}&\\ {72}& {8} \times {9\text{ ones}}\\ {240}& {8} \times {3\text{ tens}}\\ \end{aligned}$ $\begin{aligned} {2}39&\\ \underline{ \times 4{8}}&\\ {72}& {8} \times {9\text{ ones}}\\ {240}& {8} \times {3\text{ tens}}\\ {1{,}600}& {8} \times {2\text{ hundreds}}\\ \end{aligned}$ $\begin{aligned} 23{9}&\\ \underline{ \times {4}8}&\\ {72}& {8} \times {9\text{ ones}}\\ {240}& {8} \times {3\text{ tens}}\\ {1{,}600}& {8} \times {2\text{ hundreds}}\\ {360}& {40} \times {9\text{ ones}}\\ \end{aligned}$ $\begin{aligned} 2{3}9&\\ \underline{ \times {4}8}&\\ {72}& {8} \times {9\text{ ones}}\\ {240}& {8} \times {3\text{ tens}}\\ {1{,}600}& {8} \times {2\text{ hundreds}}\\ {360}& {40} \times {9\text{ ones}}\\ {1{,}200}& {40} \times {3\text{ tens}}\\ \end{aligned}$ $\begin{aligned} {2}39&\\ \underline{ \times {4}8}&\\ {72}& {8} \times {9\text{ ones}}\\ {240}& {8} \times {3\text{ tens}}\\ {1{,}600}& {8} \times {2\text{ hundreds}}\\ {360}& {40} \times {9\text{ ones}}\\ {1{,}200}& {40} \times 3\text{ tens}\\ \underline{+{8{,}000}}& {40} \times {2\text{ hundreds}}\\ \end{aligned}$ $\begin{aligned} 239&\\ \underline{ \times 48}&\\ 72}& {8} \times {9\text{ ones}}\\ 240}& {8} \times {3\text{ tens}}\\ 1{,}600}& {8} \times {2\text{ hundreds}}\\ 360}& {40} \times {9\text{ ones}}\\ 1{,}200}& {40} \times 3\text{ tens}\\ \underline{+8{,}000}}& {40} \times {2\text{ hundreds}}\\ 11{,}472 \end{aligned}$ $48 \times 239 = 11{,}472$ Let's estimate to place the decimal in $0.48\times 23.9$. $\begin{aligned} 0.48 \times 23.9 &\approx 0.5\times 20\\\\ &\approx 10 \end{aligned}$ Where can we place the decimal in $11{,}472$ to get a product close to $10$ ? $0.48\times23.9=11.472$